### Abstract

We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the BabuškaAziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained results are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of numerical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a posteriori estimates of some eigenvalues related to the error constants.

Original language | English |
---|---|

Pages (from-to) | 27-78 |

Number of pages | 52 |

Journal | Journal of Mathematical Sciences |

Volume | 17 |

Issue number | 1 |

Publication status | Published - 2010 |

### Fingerprint

### Keywords

- Error estimates
- FEM
- Interpolation error constants
- Triangular finite elements

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{0}and P

_{1}interpolations over triangular finite elements.

*Journal of Mathematical Sciences*,

*17*(1), 27-78.

**Analysis and estimation of error constants for P _{0}and P _{1} interpolations over triangular finite elements.** / Liu, Xuefeng; Kikuchi, Fumio.

Research output: Contribution to journal › Article

_{0}and P

_{1}interpolations over triangular finite elements',

*Journal of Mathematical Sciences*, vol. 17, no. 1, pp. 27-78.

_{0}and P

_{1}interpolations over triangular finite elements. Journal of Mathematical Sciences. 2010;17(1):27-78.

}

TY - JOUR

T1 - Analysis and estimation of error constants for P0and P 1 interpolations over triangular finite elements

AU - Liu, Xuefeng

AU - Kikuchi, Fumio

PY - 2010

Y1 - 2010

N2 - We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the BabuškaAziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained results are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of numerical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a posteriori estimates of some eigenvalues related to the error constants.

AB - We give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles. For the piecewise linear one, we mainly analyze the conforming case, but the present results also appear to be available for the non-conforming case. We obtain explicit relations for the upper bounds of the constants, and analyze dependence of such constants on the geometric parameters of triangles. In particular, we explicitly determine some special constants including the BabuškaAziz constant, which plays an essential role in the interpolation error estimation of the linear triangular finite element. The obtained results are expected to be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of numerical solutions based on the triangular finite elements. We also give some numerical results for the error constants and for a posteriori estimates of some eigenvalues related to the error constants.

KW - Error estimates

KW - FEM

KW - Interpolation error constants

KW - Triangular finite elements

UR - http://www.scopus.com/inward/record.url?scp=77957197530&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957197530&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 27

EP - 78

JO - Journal of Mathematical Sciences (Japan)

JF - Journal of Mathematical Sciences (Japan)

SN - 1340-5705

IS - 1

ER -